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arXiv:2610.13358

Second quantization and compactness of induced Gaussian quantum Markov semigroups

Zheng Li

math.FAquant-phSubmitted Oct 9, 2026• Announced Oct 12, 2026• 0 citations
v1 · Oct 9, 2026v2 · Oct 21, 2026
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Abstract

We characterize compactness of the Hilbert-Schmidt semigroups induced by finite-mode Gaussian quantum Markov semigroups with a faithful normal invariant state. A unitary second-quantization representation for every $ s $-embedding shows that compactness at one positive time, compact resolvent, drift stability, and uniqueness of the normal invariant state are equivalent. These conditions imply immediate trace-class membership for all embeddings. We derive exact trace-norm and Hilbert--Schmidt norm formulas, an embedding-independent trace identity, and the generator spectrum with algebraic multiplicities. Examples show that compactness does not require the existence of a spectral gap and that a spectral gap need not imply compactness when the invariant state is not unique.

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21 pages

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In arXiv:2405.04947, it was shown that a Gaussian quantum Markov semigroup on the ddd-mode bosonic Fock space with a unique faithful normal invariant state has a positive GNS spectral gap if and only if the matrix [UV‾][U \overline{V}][UV], formed from the coefficients of the Kraus operators, has 2d2d2d linearly independent columns. In this paper, we establish the corresponding criterion for the KMS spectral gap. After standardizing the invariant Gaussian state and grouping the modes according to their inverse temperatures, let UnU_nUn​ and VnV_nVn​ denote the submatrices of UUU and VVV, respectively, corresponding to the nnn-th equal-temperature block. We prove that ker⁡Un=ker⁡Vn\ker U_n=\ker V_nkerUn​=kerVn​ for every nnn, and that the KMS spectral gap is positive if and only if this common kernel is trivial for every equal-temperature block. Thus, whereas the GNS spectral gap requires a global maximal-rank condition on the Kraus coefficients, the KMS spectral gap requires maximal column rank only within each equal-temperature block.
In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be trace- less as demanded by the GKSL Theorem). Moreover, we consider the converse con- struction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corre- sponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In partic- ular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of con- tractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an or- thonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear op- erators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.